Solution of integral equations using function-valued Padé approximants II

Abstract

We consider the use of functional (i.e. function-valued) Padé approximants to accelerate the convergence of Neumann series of linear integral equations and to estimate their characteristic values and eigenfunctions.

We apply our methods to the Neumann series solution for the linear integral equation

$$\phi (x) = 1 + \lambda \int_{ - 1}^1 {K(x,y)\phi (y) dy,} $$

where

$$K(x,y) = \left\{ {\begin{array}{*{20}c} {{\textstyle{1 \over 8}}(1 + y)(1 - x), - 1 \le y \le x \le 1,} \\ {{\textstyle{1 \over 8}}(1 + x)(1 - y), - 1 \le x \le y \le 1,} \\\end{array}} \right.$$

whose kernel is non-degenerate, but whose characteristic values λ _s =(sπ)² and corresponding eigenfunctions φ _s (x)=sin1/2sπ(x+1) are well known. Estimates of λ _s and φ _s (x) derived using (hybrid) functional Padé approximants are found to be substantially more accurate than those from the Rayleigh-Ritz method, the Fredholm determinant method and the ordinary Padé approximant method.